Trace#
Definition: Trace
The trace of a square matrix \(A = (a_{ij}) \in F^{n\times n}\) is the sum of the entries on its diagonal:
\[\sum_{k=1}^n a_{k,k}\]
Notation
\[\operatorname{tr}(A)\]
Theorem: Independence of Product Order
Let \(A, B \in F^{n \times n}\) be square matrices.
The trace of the matrix product \(AB\) is the same as the trace of the matrix product \(BA\):
\[\operatorname{tr}(AB) = \operatorname{tr}(BA)\]
Proof
TODO
Theorem: Similar Matrices \(\implies\) Equal Trace
If two square matrices \(A, B \in F^{n \times n}\) are similar, then they have the same trace:
\[B = S^{-1} A S \implies \operatorname{tr}(B) = \operatorname{tr}(A)\]
Proof
TODO
Theorem: Trace and Eigenvalues
If a square matrix \(A \in F^{n \times n}\) has \(l\) distinct eigenvalues \(\lambda_1, \cdots, \lambda_l\) and the sum of their algebraic multiplicities is \(n\), then the trace of \(A\) is given as follows:
\[\operatorname{tr}(A) = \sum_{k=1}^l \lambda_k \cdot \operatorname{alg} (\lambda_k)\]
Proof
TODO